Fiberwise amenability of étale groupoids

@misc{ma2026fiberwise,
  title = {{Fiberwise amenability of étale groupoids}},
  author = {Xin Ma and Jianchao Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.09796},
  eprint = {2608.09796},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.09796},
}

TY  - UNPB
TI  - Fiberwise amenability of étale groupoids
AU  - Xin Ma
AU  - Jianchao Wu
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2608.09796
UR  - https://arxiv.org/abs/2608.09796
AB  - We introduce a new amenability property for étale groupoids, termed \textit{fiberwise amenability}, along with a stronger variant termed \emph{ubiquitous fiberwise amenability}. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on étale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for étale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a Følner--paradoxical dichotomy for minimal étale groupoids, which will serve as a key tool in a sequel on almost elementariness of étale groupoids.
ER  -